Definition 1.

The conductor of a finite abelian extension L/K is the
largest ideal CL/K⊂OK satisfying
the above properties.

Note that there is a “largest ideal” with this condition because
if proposition 1 is true for 𝒞1,𝒞2 then it
is also true for 𝒞1+𝒞2.

Definition 2.

Let I be an integral ideal of K. A ray class
field of K (modulo I) is a finite abelian extension
KI/K with the property that for any other finite
abelian extension L/K with conductor CL/K,

𝒞L/K∣ℐ⇒L⊂Kℐ

Note: It can be proved that there is a unique ray class field with
a given conductor. In words, the ray class field is the biggest
abelian extension of K with a given conductor (although the
conductor of Kℐ does not necessarily equal
ℐ !, see example 2).

if and only if 𝔭 splits completely in L. Thus we obtain a characterization of the ray class field of conductor 𝒞 as the abelian extension of K such that a prime of K splits completely if and only if it is of the form